Somehow, I suspect I wouldn’t survive long on the frontier.
Drop me in the American West, circa 1850, and I fear my math-blogging and bad-drawing skills might not carry me far. I need indoor plumbing. I need the rule of law. I need chain coffee shops. I’m not cut out for the frontier.
And yet the frontier is exactly where I found myself the other day, when I came across this formula in the wonderful Penguin Book of Curious and Interesting Numbers, by David Wells:
I decided to play around with this product a bit. After all, what are products for, if not playing around?
(Go ahead and play with your Apple products. I’ll play with my infinite ones. We’ll see who has more fun.)
I felt like there should be an easier way to write this expression, exploiting the repetition of factors, so I gave it a shot, and created this:
Then my brain exploded and the universe dissolved around me, because I had just punched logic in the face, and it had punched me back.
The left side of that equation is π/2. It’s roughly 1.57.
The right side of that equation, however, is a product of many numbers—all of them below 1.
What happens when you multiply two numbers smaller than 1? You get another number smaller than 1.
How the heck could that equal 1.57?
Somehow, law had broken down. I was stranded on the frontier, with no cavalry or sheriff to come save me. This was a dispute I’d have to resolve myself, by hook, by crook, or by showdown at high noon.
At this point, I faced three grim possibilities:
- The cool book where I found this formula was wrong.
- Logic was wrong.
- I was wrong.
Pitting my own abilities against (1) those of David Wells and (2) the tensile strength of the fabric of the cosmos, it became pretty clear where the error lay. So I set out to find where I’d gone wrong.
Going back to the original equation, I came up with an alternative way to rewrite it:
This one was clearly greater than 1. (After all, it was an infinitely long product, with each of its factors greater than 1.) Checking a few terms in Excel verified that it was approaching π/2.
Then I found another way to rewrite the original:
This one starts at 2. Then it gets smaller and smaller, by tinier and tinier adjustments. Once again, it seemed to approach π/2.
So what had I done wrong in my original approach?
Soon enough I realized it. I’d made a mistake that seemed perfectly innocuous at the time but had, in fact, contained the seed of my own annihilation. I’d made a move that doesn’t matter with finite products, but which is forbidden with infinite products.
I’d forgotten to multiply by 1.
It perhaps goes without saying, but when you’re working with infinities, you need to be careful. Notation that’s usually harmless can explode like dynamite, making rubble of everything.
Infinity is the frontier. Law breaks down.
Back home, in the realm of finite products, multiplying by 1 makes no difference. 3 x 2 is the same as 3 x 2 x 1 (or as 1 x 3 x 2 x 1 x 1, for that matter). Finite products—that is, lists of factors that eventually end—can be rearranged to your heart’s content.
Not so with infinite products. Just look what else I can do, if I allow myself to play around with 1’s:
Throwing in extra 1’s allows me to rearrange the numerators and denominators virtually however I please. It seems I can make this product as large or as small as I want!
I should also confess that steps like the second one above – where I take a long product of fractions, and turn it into a single fraction with a long product on top and another long product on bottom – are a little dubious in this setting. With finite products? Fine. With infinite ones? Not so much.
In short: the frontier is a strange place.
I’m working here with products, but the same dangers apply with sums. Take this one, called the alternating harmonic series:
This sequence starts at 1.
Then it jumps backwards to ½.
Then it jumps forwards to 5/6.
Then it jumps backwards to 7/12.
Then it jumps forwards to 47/60.
As you go, the sum keeps jumping forwards and backwards, forwards and backwards… but by smaller and smaller steps. Looking carefully, you can see that it’s hovering around a certain destination—a place it will never in our lifetimes reach, but which it approaches.
That point is smaller than 1.
It’s bigger than ½.
It’s smaller than 5/6.
It’s bigger than 7/12.
In fact, it turns out to be roughly 0.693, or more precisely, ln(2).
So far, so good. But what if I rearrange the terms, like this?
We start at 1.
Then we add something positive.
Then we add something else positive.
And so on.
Clearly, the answer has to be larger than 1—which 0.693 is decidedly not. So I’ve changed the result sum, simply by rearranging the terms.
That’s not supposed to happen with addition! 3 + 4 is the same as 4 + 3. Making your family members stand in a different order shouldn’t make your family bigger, smaller, smarter, or taller—it’s the same family, isn’t it?
But with infinite things to add, the order in which you arrange them turns out to matter a great deal. In fact, this sequence can be rearranged to form literally any number you like.
Why do we explore this mathematical frontier? What drives us to abandon the safe coastal cities that we’ve always known, and throw ourselves into the unknown interior of the mathematical continent? Aren’t we afraid? Mightn’t we get eaten by bears, swallowed by rapids, or tangled in paradoxes? Why do we risk it?
Simple: because humans are explorers.
It’s as true of mathematicians as it is of Lewis and Clarke. We want to chart new landscapes, to find fresh challenges, to go where our old ways don’t necessarily help or hold true. We enjoy the thrill of surviving on our wits alone.